Blow-up of Radially Symmetric Solutions of a Non-local Problem Modelling Ohmic Heating
| dc.contributor.author | Tzanetis, Dimitrios E. | |
| dc.date.accessioned | 2020-07-13T20:01:36Z | |
| dc.date.available | 2020-07-13T20:01:36Z | |
| dc.date.issued | 2002-02-01 | |
| dc.description.abstract | We consider a non-local initial boundary-value problem for the equation ut = ∆u + λƒ(u) / (∫Ω ƒ(u) dx)2, x ∈ Ω ⊂ ℝ2, t > 0, where u represents a temperature and ƒ is a positive and decreasing function. It is shown that for the radically symmetric case, if ∫∞0 ƒ(s) ds < ∞ then there exists a critical value λ* > 0 such that for λ < λ* there is no stationary solution and u blows up, whereas for λ < λ* there exists at least one stationary solution. Moreover, for the Dirichlet problem with -s ƒ'(s) < ƒ(s) there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if λ < λ* then there are at least two solutions, which if λ = λ* at least one solution. Stability and blow-up of these solutions are examined in this article. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 26 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Tzanetis, D. E. (2002). Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. <i>Electronic Journal of Differential Equations, 2002</i>(11), pp. 1-26. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/12051 | |
| dc.language.iso | en | |
| dc.publisher | Southwest Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2002, San Marcos, Texas: Southwest Texas State University and University of North Texas. | |
| dc.subject | Nonlocal parabolic equations | |
| dc.subject | Blow-up | |
| dc.subject | Global existence | |
| dc.subject | Steady states | |
| dc.title | Blow-up of Radially Symmetric Solutions of a Non-local Problem Modelling Ohmic Heating | |
| dc.type | Article |